basis-map coefficients of A371080, shifted → basis A013610 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-25 + 30\,x - 9\,x^{2} + 30\,n - 18\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 6\,x + 6\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	6							7
2	4	30	27						61
3	28	234	324	108					694
4	280	2508	4293	2376	405				9862
5	3640	34284	67095	48060	14175	1458			168712
6	58240	570384	1227798	1037340	407025	74358	5103		3380248
7	1106560	11186736	25894890	24620652	11623500	2878092	357210	17496	77685136

All 15 rows

Technical details

Node id: n-7a174e012bdbab42 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-5e4d568b2cdb4901

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         1,          6;
  2:         4,         30,          27;
  3:        28,        234,         324,         108;
  4:       280,       2508,        4293,        2376,         405;
  5:      3640,      34284,       67095,       48060,       14175,       1458;
  6:     58240,     570384,     1227798,     1037340,      407025,      74358,      5103;
  7:   1106560,   11186736,    25894890,    24620652,    11623500,    2878092,    357210,     17496;
  8:  24344320,  252747552,   620027892,   645233904,   348044445,  105162624,  17931942,   1609632,   59049;
  9: 608608000, 6464754720, 16638061284, 18610959168, 11120738265, 3882025602, 816367734, 101721744, 6908733, 196830;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(18*n-30)*x+9*x^2)*P(n-2,x) + (6*n-7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*P[n - 2, x] + (6*n - 7 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*v[i - 2] + (6*n - 7 + 6*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.